The only value of $x$ for which ${2^{\sin x}} + {2^{\cos x}} > {2^{1 - (1/\sqrt 2 )}}$ holds, is

Number of solutions of the equation $2^x + x = 2^{sin \ x} +  \sin x$ in $[0,10\pi ]$ is -

The number of solutions of the equation $\sqrt[3]{{\sin \theta  - 1}} + \sqrt[3]{{\sin \theta }} + \sqrt[3]{{\sin \theta  + 1}} = 0$ in $[0,4\pi]$ is 

If $\tan m\theta = \tan n\theta $, then the general value of $\theta $ will be in

Let $S=\left\{\theta \in[-\pi, \pi]-\left\{\pm \frac{\pi}{2}\right\}: \sin \theta \tan \theta+\tan \theta=\sin 2 \theta\right\} \text {. }$  If $T =\sum_{\theta \in S } \cos 2 \theta$, then $T + n ( S )$ is equal

If $\sin x=\frac{3}{5}, \cos y=-\frac{12}{13},$ where $x$ and $y$ both lie in second quadrant, find the value of $\sin (x+y)$.